Recall that there is an equivalence of categories (Dold-Kan) $$N:\mathrm{s}\mathbf{Ab}\simeq \operatorname{Ch}_{\geq 0}(\mathbf{Ab}):\Gamma$$ between simplicial abelian groups and (connective) chain complexes, where $N$ sends a simplicial abelian group to its associated normalized chain complex.
Using this equivalence of categories, we can, by transport of structure, give an unorthodox tensor product on the category of chain complexes. We may define this by the formula $X\otimes_\Delta Y=N(\Gamma(X)\otimes \Gamma(Y))$.Gamma(Y)),$ where the tensor product on the righthand side is the tensor product (taken pointwise) of simplicial abelian groups.
Then my question: Is there an explicit description of this tensor product in terms of the chain complexes themselves?
